Prove Geometric Sequence with $(a,b,c)$

In summary, a geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a fixed number, called the common ratio. To prove that a sequence is geometric, one must show that each term is found by multiplying the previous term by the common ratio. This can be done by finding the ratio between any two consecutive terms and showing that it is consistent throughout the sequence. The formula for the nth term in a geometric sequence is an = a * r^(n-1), where a is the first term and r is the common ratio. A geometric sequence can also have a negative common ratio, but it must have an absolute value less than 1 for the sequence to be convergent.
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Let $a,\,b,\,c$ be non-zero real numbers such that $(ab+bc+ca)^3=abc(a+b+c)^3$. Prove that $a,\,b,\,c$ are terms of a geometric sequence.
 
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Consider the monic polynomial $P(x)=x^3+mx^2+nx+p$, with roots $a,\,b,\,c$. Then by Viete's relations we have

$a+b+c=-m\\ab+bc+ca=n\\abc=-p$

The given equality yields $n^3=m^3p$. Hence, if $m\ne 0$, the equation $P(x)=0$ can be written as

$x^3+mx^2+nx+\dfrac{n^3}{m^3}=0$

$m^3x^3+m^4x^2+nm^3x+n^3=0$

Factor the left-hand side,

$(mx+n)(m^2x^2-mnx+n^2)+m^3x(mx+n)=(mx+n)(m^2x^2+(m^3-mn)x+n^2)$

It follows that one of the roots of $P$ is $x_1=-\dfrac{n}{m}$ and the other two satisfy the condition $x_2x_3=\dfrac{n^2}{m^2}$. We obtained $x_1^2=x_2x_3$, thus the roots are terms of a geometric sequence.

If $m=0$ then $n=0$ but in this case, the polynomial $x^3+p$ cannot have three real roots.
 

FAQ: Prove Geometric Sequence with $(a,b,c)$

1. What is a geometric sequence?

A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a constant value, known as the common ratio. The general form of a geometric sequence is a, ar, ar^2, ar^3, ... where a is the first term and r is the common ratio.

2. How do you prove a geometric sequence with $(a,b,c)$?

To prove a geometric sequence with $(a,b,c)$, you need to show that the ratio between any two consecutive terms is constant. This can be done by dividing the second term by the first term, and then dividing the third term by the second term. If the resulting values are equal, then the sequence is a geometric sequence with a common ratio of b/a. You can continue this process for all terms in the sequence to further support your proof.

3. What are some real-life examples of geometric sequences?

Geometric sequences can be found in various natural phenomena, such as the growth of bacteria, population growth, and the spread of diseases. They are also commonly used in financial calculations, such as compound interest and depreciation of assets.

4. Can a geometric sequence have negative terms?

Yes, a geometric sequence can have negative terms. As long as the common ratio is a negative number, the sequence will alternate between positive and negative terms.

5. How is a geometric sequence different from an arithmetic sequence?

A geometric sequence differs from an arithmetic sequence in that the terms are found by multiplying the previous term by a constant value, whereas in an arithmetic sequence, the terms are found by adding a constant value to the previous term. Additionally, in a geometric sequence, the difference between any two consecutive terms is not constant, unlike in an arithmetic sequence.

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